PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

An adjoint triple induces adjunctions between its associated endofunctors

Statement

If $L ⊣ M ⊣ R$ , with $L, R : D \to C$ and $M : C \to D$ , then

$L M ⊣ R M on C,$

and

$M L ⊣ M R on D .$

These adjunctions use the composite units and counits and require no local-smallness hypothesis.

Facts & Assumptions

Given: An adjoint triple $L ⊣ M ⊣ R$ .

[L1]

An adjoint triple supplies adjunctions $L ⊣ M$ and $M ⊣ R$ with the displayed functor types (Adjoint triple $L ⊣ M ⊣ R$ ).

[L2]

If $F ⊣ G$ and $F^{'} ⊣ G^{'}$ , then $F^{'} F ⊣ G G^{'}$ with the composite unit and counit formulas (Adjunctions compose with the composite unit and counit formulas).

Proof

technique · direct

1.1L1L2

Compose $M ⊣ R$ with $L ⊣ M$ in [L2]. The left composite is $L M : C \to C$ and the right composite is $R M : C \to C$ , so $L M ⊣ R M$ .

1.2L1L2

Compose $L ⊣ M$ with $M ⊣ R$ in [L2]. The left composite is $M L : D \to D$ and the right composite is $M R : D \to D$ , so $M L ⊣ M R$ .

2.1step 1.1step 1.2L2∎

In each case the unit is obtained by inserting the second unit between the first unit's functors, and the counit by inserting the first counit between the second counit's functors, exactly as in [L2]. Since [L2] uses unit-counit data, neither construction requires hom-sets.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Emily Riehl, Category Theory in Context, 2nd ed., Exercise 4.1.iv (standard reference, not scraped)